The multiscale hybrid-hybrid-mixed method MH2M - Barros, Franklin da Conceição de (2022)
Many problems of practical interest in science and engineering are of the multiscale type, and many of those can be described by partial differential equations (PDEs) with oscillatory coefficients. The corresponding numerical solutions using classical methods are extremely expensive in terms of memory and CPU. Multiscale schemes such as the MsFEM and MHM methods have been developed to solve such problems, based on two-level ideas. This work proposes a numerical method, the Multiscale Hybrid-Hybrid-Mixed method (MH2M). The starting point is a hybrid formulation of three fields: the solution in each element interior, its flux at the boundary of each element, and its trace on the mesh skeleton. Continuity of traces and fluxes are weakly imposed. Multiscale effects are incorporated into basis functions through localized Neuman problems. A series of static condensations transforms the saddle point problem into an elliptic one, posed at the interfaces. At the discrete level, this drastically reduces the size of the global system. The matrix of the associated linear system is symmetric and positive definite, and can be solved by classical iterative schemes. We prove the well-posedness of the method and establish error estimates. We also perform numerical tests to confirm the theoretical predictions and compare the method with the FEM, MsFEM and MHM schemes.